Will this help program my 19 button mouse?
The "Programmable buttons" are usually configured in a system control panel for the device, you should find a place where you can specify what action/key is assigned to each button.

Another approach to try is to open blenders input preferences and adjust an action you want to assign to a button. Where the mouse/key combination is shown, you can click that and the next mouse or keypress will be assigned to the action. Try adjusting an action and pressing one of the mouse buttons to see if blender recognises the button, if not you will need to rely on the system configuration to set each button to emulate keystrokes.

While I am not familiar with the addon you mention, blender is very customisable, you can setup the shortcuts to suit what works for you, modifier keys can be disabled so that menus are displayed with single key presses, using python you can create your own menus - either a vertical list of items or a pie menu.